John and Jivanti Together Have 45 Marbles
Question:
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Form the quadratic equation to find how many marbles they had to start with, if John had \(x\) marbles.
Solution
Let John originally have
\[ x \]
marbles.
Since together they have 45 marbles, Jivanti originally had
\[ 45-x \]
marbles.
After each loses 5 marbles:
John has \[ x-5 \] marbles.
Jivanti has \[ 45-x-5=40-x \] marbles.
According to the question,
\[ (x-5)(40-x)=124 \]
Expanding,
\[ 40x-x^2-200+5x=124 \]
\[ -x^2+45x-200=124 \]
\[ -x^2+45x-324=0 \]
Multiplying by \(-1\),
\[ x^2-45x+324=0 \]
Required Quadratic Equation
\[ \boxed{x^2-45x+324=0} \]
Answer
If John originally had \(x\) marbles, then the required quadratic equation is
\[ \boxed{x^2-45x+324=0} \]
This equation can be solved to determine the number of marbles John and Jivanti had initially.